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In 3 dimensions, there are 8 geometries. The geometrization proved by Grigori Perelman states that every 3-manifold can be cut into pieces that are geometrizable.

Three-manifold with finite fundamental group are spherical.

etale homotopy of ${\mathcal{O}}_K$ for $K$ a Global field : a punctured 3-manifold

Theorem: Every compact Three-manifold arises as a combination of (2 different variants of) Weinstein surgeries on $S^3$.